Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups
aa r X i v : . [ m a t h . N T ] D ec Computation of Hurwitz spaces and new explicit polynomials foralmost simple Galois groups
Joachim K¨onig
Universit¨at W¨urzburg, Emil-Fischer-Str. 30, 97074 W¨urzburg, Germany
Abstract
We compute the first explicit polynomials with Galois groups G = P Γ L (4), P GL (4), P SL (4)and P SL (2) over Q ( t ). Furthermore we compute the first examples of totally real polynomialswith Galois groups P GL (11), P SL (3), M and Aut ( M ) over Q . All these examples makeuse of families of covers of the projective line ramified over four or more points, and therefore usetechniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g.by Malle ([23]), Couveignes ([4]), Granboulan ([12]) and Hallouin ([13]). Unlike previous examples,however, some of our computations show the existence of rational points on Hurwitz spaces thatwould not have been obvious from theoretical arguments. Keywords:
Galois theory; polynomials; moduli spaces; symbolic computation
1. Introduction
In recent years, there has been a great deal of progress in explicit computation of polynomialswith prescribed Galois group. One notable area of interest is the computation of 3-point coversof the line (Belyi maps), for which strong tools have been developed, e.g. in [17]. Such techniqueshave been used to calculate explicit polynomials for many permutation groups of small degrees.Often the existence of such polynomials defined over Q could a priori be deduced by the RigidityMethod (cf. [26, Chapter I]). However, even for almost simple groups of relatively small degree, notall questions can be answered merely via 3-point covers.Meanwhile, covers with more than three branch points have been computed to solve some ofthose problems, like finding totally real polynomials with given Galois group, but also because they Email address: [email protected] (Joachim K¨onig)
Preprint submitted to Elsevier
INTRODUCTION G degree § P SL (2) 31 4 P α no § P Γ L (4) 21 4 P α no P GL (4) 21 4 no derived from previous;base curve not generi-cally P P SL (4) 21 5 no see previous § P Γ L (4) 21 4 P α yes § P GL (11) 22 4 rank 1 ell. curve yes P SL (11) 11 4 yes § P GL (11) 12 4 rank 1 ell. curve yes P SL (11) 11 5 yes § P SL (3) 13 5 P α,β yes § Aut ( M ) 22 4 rank 1 ell. curve yes M
22 4 yes derived from previous;base curve P Table 1: Overview of the polynomials computed in this article sometimes give rise to multi-parameter polynomials over Q . A spectacular result in the computationof covers with more than three branch points was Granboulan’s explicit M -polynomial in [12].An important source for examples of multi-parameter polynomials is Malle’s paper [23], which also,along with Couveignes’ [4] and [5], outlines methods for their calculation.The computational results of this article can be largely divided into two areas: the calculation ofexplicit polynomials for some of the almost simple groups of smallest permutation degree for whichno polynomials were previously known; and the calculation of the first totally real polynomials forother almost simple groups.Table 1 summarizes very briefly the basic features of the families of polynomials occurringand of the Hurwitz spaces that they are parametrized by. Here, a Hurwitz variety P α leads toa one-dimensional family of covers, parameterized by α , of the projective line P t , and thereforea two-parameter polynomial f ( α, t, x ) ∈ Q ( α, t )[ x ] with the prescribed Galois group. Similarly,the Hurwitz variety P α,β in Section 7 leads to a three-parameter polynomial. In the “elliptic-curve” cases, one obtains the existence of an infinite family f P ( t, x ) of one-parameter polynomials,parameterized by the rational points P of a rank-1 elliptic curve; for the sake of simplicity, onlysample polynomials of these families are given. In some cases, polynomials for normal subgroupsare derived in a natural way from polynomials with a given group. In these cases, it is to beunderstood in Table 1 that the Hurwitz variety parameterizing the family of covers is the same as THEORETICAL BACKGROUND
2. Theoretical background
We recall some basic facts about monodromy of covers, Hurwitz spaces and braid group action.For a deeper introduction, cf. [10], [28] or [31].
Let S = { p , ..., p r } be a finite subset of the projective line P C , p ∈ P C \ S , and f : R → P C \ S an n -fold covering map. Then the fundamental group π ( P C \ S, p ) acts on the fiber f − ( p ) vialifting of paths. This yields a homomorphism of π ( P C \ S, p ) into S n , and if γ i are homotopyclasses of closed paths from p around p i ( i = 1 , ..., r ), ordered counter-clockwise in P C , their imagesunder this action, say σ , ..., σ r , generate a group isomorphic to the Galois group of E | C ( t ), with E being the Galois closure of the function field of (the compact Riemann surface) R . Furthermore,we have σ · · · σ r = 1. We call ( σ , ..., σ r ) the branch cycle description of the cover f . The genus g of R is given by the Riemann-Hurwitz genus formula g = − ( n −
1) + 12 r X i =1 ind ( σ i ) , where the index ind ( σ i ) is defined as n minus the number of cycles of σ i ∈ S n . This motivates thefollowing definition: Definition 1 (Genus- g tuple) . Let G ≤ S n be a transitive permutation group, r ∈ N and σ , ..., σ r ∈ G such that h σ , ..., σ r i = G and σ · · · σ r = 1. Then ( σ , ..., σ r ) is called a genus- g tuple of G , with g := − ( n −
1) + P ri =1 ind ( σ i ). Let G be a finite group. Let S be a subset of the projective line P C of cardinality r , p beany point in P \ S and f : π ( P \ S, p ) → G be an epimorphism mapping none of the canonical THEORETICAL BACKGROUND γ , ..., γ r of the fundamental group to the identity. On the set of such triples ( S, p , f )one defines an equivalence relation via ( S, p , f ) ∼ ( S ′ , p ′ , f ′ ) : ⇔ S = S ′ and there exists a path γ from p to p ′ in P \ S such that the induced map γ ⋆ : π ( P \ S, p ) → π ( P \ S, p ′ ) on thefundamental groups fulfills f ′ ◦ γ ⋆ = f . Identifying the group G with the deck transformation groupof a Galois cover ϕ : X → P \ S , Riemann’s existence theorem leads to a natural identification ofthese equivalence classes [ S, p , f ] with equivalence classes [ ϕ, h ], where ϕ : X → P \ S is a Galoiscover that can be extended to a branched cover of P with exactly r branch points, and h is anisomorphism from the group of deck transformations of ϕ to G . Cf. [10, Section 1.2.] (especiallyfor the precise identification between the two different sets of equivalence classes) and [31, 10.1].Denote the set of these equivalence classes by H inr ( G ). This space carries a natural topologicalstructure, and also the structure of an algebraic variety. This directly links the inverse Galoisproblem with the existence of rational points on certain algebraic varieties. The main result is thefollowing (cf. [31, Cor. 10.25] and [8, Th. 4.3]): Theorem 1.
Let G be a finite group with Z ( G ) = 1 . There is a universal family of ramifiedcoverings F : T r ( G ) → H inr ( G ) × P C , such that for each h ∈ H inr ( G ) , the fiber cover F − ( h ) → P C is a ramified Galois cover with group G . This cover is defined regularly over a field K ⊆ C if andonly if h is a K -rational point. In particular, the group G occurs regularly as a Galois group over Q if and only if H inr ( G ) has a rational point for some r . Monodromy action leads to a group theoretic interpretation of the above equivalence classes ofcovers.
Definition 2 (Nielsen class) . Let G be a finite group, r ≥ E r ( G ) := { ( σ , ..., σ r ) ∈ ( G \ { } ) r | σ · ... · σ r = 1 , h σ , ..., σ r i = G } the set of all generating r -tuples in G \ { } with product 1. Furthermore let E inr ( G ) be the quotientof E r ( G ) modulo conjugating the tuples simultaneously with elements of G .For any r -tuple C := ( C , ..., C r ) of non-trivial conjugacy classes of G the Nielsen class N i ( C )is defined as the set of all ( σ , ..., σ r ) ∈ E r ( G ) such that for some permutation π ∈ S r it holds that σ i ∈ C π ( i ) for all i ∈ { , ..., r } . The definition of N i in ( C ) is then possible in analogy to the abovenotation.Denote by H r the Hurwitz braid group on r strands. This group, a quotient of the Artin braidgroup, can be defined as the group generated by r − β , ..., β r − fulfilling the classicalbraid relations β i β j = β j β i for 1 ≤ i < j − ≤ r − ,β i β i +1 β i = β i +1 β i β i +1 for 1 ≤ i ≤ r − THEORETICAL BACKGROUND β · · · β r − β r − · · · β = 1(cf. [26, Chapter III.1.1 and III.1.2]). The group H r acts naturally on the set E r ( G ) (with aninduced action on E inr ( G )) via( σ , ..., σ r ) β i := ( σ , ..., σ i − , σ i σ i +1 σ − i , σ i , ..., σ r ) , for i = 1 , ..., r − . (1)It is obvious that the sets N i in ( C ) are unions of orbits under these actions.Furthermore, if U r denotes the space of r -sets in P C and Ψ : H in ( G ) → U r is the branchpoint reference map, the elements of a given fiber are in 1-1 correspondence with elements of E inr ( G ). Indeed, the above action on equivalence classes of r -tuples of elements of G is, via thiscorrespondence, essentially the same as the action of the fundamental group on the fiber via liftingof paths. Each of the orbits of the braid group acting on N i in ( C ) corresponds to a connectedcomponent of H inr ( G ). The union of all connected components corresponding to N i in ( C ) is whatis usually referred to as a Hurwitz space: Definition 3 (Hurwitz spaces) . For an r -tuple C of conjugacy classes of a group G with a non-empty Nielsen class N i in ( C ), the union of components of H inr ( G ) corresponding to N i in ( C ) is calledthe (inner) Hurwitz space of C .If one leaves out the permutation π in the above definition of a Nielsen class, one gets the notionof a straight Nielsen class: SN i ( C ) := { ( σ , ..., σ r ) ∈ E r ( G ) | σ i ∈ C ( i ) for i = 1 , ..., r } The definition of
SN i in ( C ) is then possible in analogy to Def. 2.Now always assume that Z ( G ) = { } , and that the braid group action on SN i in ( C ) is tran-sitive. Following [8, Theorem 4.3], one has the following morphisms between (quasi-projective)varieties: • F : T → H in ( C ) × P , the universal family of covers in the Nielsen class N i in ( C ). • H in ( C ) → U r , mapping each point of H in ( C ) to its set of branch points. This condition assures that the Hurwitz space is an absolutely irreducible variety over its field of definition. Buteven in the case of intransitive braid group action, there may still be an absolutely irreducible component, grantedthat there is a “rigid” braid orbit, e.g. a unique orbit of a given length.
COMPUTATIONAL METHODS • Proceeding to the pullback ( H in ) ′ ( C ) := H in ( C ) × U r U r , one also obtains a morphism( H in ) ′ ( C ) → U r , with U r the space of ordered r -sets in P C . • Via
P GL -action, ( H in ) ′ ( C ) is birationally equivalent to P C × P C × P C × H red ( C ), where H red ( C ) is the image under the above map of the subvariety of ( H in ) ′ ( C ) consisting of coverswith the first three branch points equal to 0, 1, and ∞ (in this order). • This restriction gives a morphism of r − H red ( C ) → U r − .Particularly in the case r = 4, C := H red ( C ) is a curve - it corresponds, via action of P GL ( C ), tothe set of all covers with branch cycle description in C and ordered branch point set (0 , , ∞ , λ ),for some λ ∈ C \ { , } (Of course, this choice of branch points cannot always be assumed for coversdefined over Q ; therefore one may consider covers with partially symmetrized branch point sets aswell - cf. Chapter III.7 in [26]). The existence of Galois covers defined over a field K is thereforedirectly linked to the existence of K -points on such curves (often called reduced Hurwitz spaces).We also refer to these reduced Hurwitz spaces as Hurwitz curves. There are well known theoreticalcriteria to determine the genus of these Hurwitz curves, cf. e.g. Thm. III.7.8 in [26].
3. Computational methods
Let
N i ( C ) be a Nielsen class of genus zero 4-tuples generating a finite group G (assume always Z ( G ) = { } ). Recall from Section 2 that, if SN i in ( C ) contains a unique orbit of length n underthe action of the braid group, H is the corresponding connected component of the (inner) Hurwitzspace and H ′ its pullback over U , then there is a natural degree- n cover H ′ → U , where H ′ isbirationally equivalent to C × ( P C ) , and a degree- n cover C → P C of (irreducible projectivenon-singular) curves. If, via Moebius transformations, one fixes three of the four branch points ofthe genus zero covers, say to 0, 1 and ∞ , one obtains a family of branched covers T → C × P C .Let t be a parameter for the projective line on the right side, then this family will have orderedramification locus in t : (0 , λ, , ∞ ), where λ is a function on C . As C is an irreducible curve, itsfunction field is of one variable (and of degree n over C ( λ )), i.e. equal to C ( λ, α ) for some function α . Therefore the family T → C × P C can be expressed by a polynomial equation f ( λ, α, t, X ) = 0,where f ∈ C ( λ, α )[ t, X ] is linear in t (because of the genus zero condition). For every specialization t t (e.g. to a ramification point), the coefficients of f ( λ, α, t , X ) lie in the function field C ( λ, α ). COMPUTATIONAL METHODS C ( λ ) into the Laurent series field C (( λ )). Then, using thefact that the finite extensions of C (( λ )) are all equal to some C (( µ )) with µ e = λ , for some e ∈ N (cf.[31, Chapter 2.1.3]), all of these coefficients have a Puiseux expansion in λ , i.e. can be written as aLaurent series in µ := λ e with some e ∈ N . Here the exponent e is nothing but the ramification indexin the Hurwitz space of some place lying over λ
0. This ramification index can be determinedby group theoretical means: it is the number of equivalence classes of covers, i.e. of equivalenceclasses of 4-tuples ( σ , σ , σ , σ ) in SN i in ( C ), that lead to the same degenerate cover, i.e. classtriple ( σ σ , σ , σ ), upon letting λ converge to zero.There are two important cases for practical computations: • If one knows an explicit polynomial for some degenerate (3-point) cover with monodromy( σ σ , σ , σ ) as above, one can determine e and then develop Puiseux expansions to regain acover with 4 branch points. The idea is to gain a sufficiently good initial approximation andthen use Newton iteration to develop the series. A point in a given fiber of the non-degeneratecover which converges to a multiplicity- k point X x of the degenerate cover will be of theform X x + O ( µ e/k ). To reach the necessary precision of the initial approximation, oneneeds to determine the unknown first-order coefficient. This is achieved by finding equationsfor the “opposite” degeneration with monodromy ( σ , σ , σ σ ), corresponding to µ → s := t/λ .A detailed description of this method has been given by Couveignes in [4], and an explicitMagma algorithm is contained in [20]. Compare also the examples in the later sections,especially Section 4.3. • If one even knows an explicit polynomial for some non-degenerate (4-point) cover of the family(say, ramified in t (0 , , ∞ , a ) for some a ∈ C \ { , } ), then by mapping the branch pointsof the family to t (0 , , ∞ , a + λ ) one can develop from an unramified point, i.e. actuallyobtain Laurent series in λ for the above coefficients. As one starts from a non-ramified pointon the Hurwitz space, there is also no concern of getting into the Hurwitz space of a wrongfour-point family by deforming, so computations can be done modulo suitable primes (as onedoesn’t need to double-check the monodromy via numerical methods in C ). However, forgroups of larger degree, one cannot expect to directly find a polynomial for a non-degeneratecover, as the corresponding system of equations becomes too complicated. COMPUTATIONAL METHODS Remark: a) Of course all this remains true for r -tuples with r ≥ r − t (in the unsymmetrized case).b) So far, all considerations were made over C . However, for suitable choice of the conjugacyclasses in N i ( C ), the corresponding Hurwitz space can sometimes be defined over Q . ThePuiseux expansion approach may therefore be carried out over an appropriate number field.c) The above condition on the ordered ramification locus in t to be t (0 , , ∞ , λ ) correspondsto the unsymmetrized case; analogously, suitable Moebius transformations lead to differentsymmetrized cases; e.g. in the C -symmetrized case one can w.l.o.g. consider all covers withordered ramification locus ( { zeroes of t − λ } , , ∞ ), etc. Assume for simplicity that the reduced Hurwitz space (obtained from H in ( C ) via P GL -action)for a given family of covers with r branch points can be defined over Q . As this reduced Hurwitzspace is an ( r − r − r − Q . In particular, the coefficients of an equation f ( t, x ) = 0 for the corresponding universal family ofcovers (cf. the following sections) are such elements. This enables one to obtain explicit equationsdefining the Hurwitz space over Q .Again, for sake of simplicity, assume r = 4, then the function field extension corresponding tothe reduced Hurwitz space cover is of the form F := Q ( λ, α ) | Q ( λ ), with a function field F of onevariable. The Puiseux expansion approach has embedded F into the Laurent series field K (( λ /e ))(for a suitable e ∈ N and a suitable number field K ). There are now different ways to obtaindependencies between two coefficients α , α of the model. Under certain additional conditions, itwill be clear that Q ( α , α ) is already the full function field F and therefore the algebraic dependencybetween α and α is actually a defining equation for the Hurwitz curve. E.g., if the braid groupacts primitively on the given Nielsen class, then there is no intermediate field between F and Q ( λ ),so α := λ and α any coefficient not contained in Q ( λ ) will suffice. This is usually not the best Otherwise one gets the analogous results over some number field K . COMPUTATIONAL METHODS F : Q ( λ )] = | SN i in ( C ) | is often considerably larger than some other degrees [ F : Q ( α i )](see the next section for theoretical results on the gonality of F ).The following approaches will be used in the following sections to obtain algebraic dependencies(cf. also Section 5 of [4]):1) If the coefficients α i are actually given as Laurent series in µ := λ /e , simply solve a systemof linear equations in order to see whether α , α fulfill a polynomial equation of degrees n , n respectively. As such an equation has N := ( n + 1)( n + 1) unknowns, series needto be expanded to precision at least µ N in order to obtain sufficiently many equations viacomparison of coefficients.An explicit (and precise!) Laurent series expansion is usually difficult to obtain over Q , asthe coefficients grow quite rapidly. Therefore this approach, at least for dependencies of highdegrees, can often be only obtained modulo some prime.2) Once the degrees for algebraic dependencies are known (or can be conjectured, e.g. after mod- p reduction), the corresponding systems of linear equations can also be solved numerically forcomplex approximations, with many different specialized values for λ , instead of one high-order Laurent series in λ .3) Instead of solving approximate complex equations numerically, a mod- p solution can be liftedto many different solutions in Z p . The algebraic dependencies can then be retrieved viainterpolation.4) If the degrees are not too high, algebraic dependencies can be obtained from complex approx-imations via the LLL-algorithm (see [21]): suppose that α , α fulfill a rational polynomialequation of degrees n and n respectively, specializing α to a rational value will leave α ina number field of degree at most n over Q . With sufficient precision, we managed to retrievethe minimal polynomials for these specialized values of α for degrees n up to 100. Again,repeating this for many (at least n +1) different specializations for α will allow interpolationto retrieve the original equation. Remark:
Especially for larger braid orbits, with braid genus g >
0, it may not always be possible to directlyfind algebraic dependencies for all coefficients occurring in an equation for the universal family (as
COMPUTATIONAL METHODS U r ) in order to gain, from an approximation for a cover with branch cycle description ( σ , ..., σ r ),approximations for all covers with the same set of branch points and branch cycle description inthe same braid orbit. E.g., applying the braid β i to a given cover with ordered branch point set( p , ..., p r ) corresponds to switching the i -th and the ( i + 1)-th branch point by moving each of themby 180 degree on the the disc around p i + p i +1 with radius | p i − p i +1 | (assuming this disk contains noother branch points). See [31, Lemma 10.9]. Usually the algebraic dependencies f ( a, b ) = 0 will not be optimal with regard to the degrees ofthe variables a, b involved. One can therefore use considerations about the gonality of the functionfield K ( a, b ), involving computations of Riemann-Roch spaces, to find good parameters, i.e. rationalfunction fields with low index in the function field K ( a, b ). This is especially useful in function fieldsof genus 0 or 1, or in hyperelliptic function fields. Definition 4 (Gonality) . Let F | K be a function field of one variable. The gonality gon ( F | K ) of F | K is defined as the minimum of the degree [ F : K ( x )] (for x ∈ F ), i.e. the minimal index of arational function field in F .We use the following estimates on the gonality of function fields, which also yield a method toexplicitly find rational function fields K ( x ) ⊆ F of low index. Lemma 2.
Let g be the genus of the function field F | K . Thena) If g = 0 , then gon ( F | K ) ≤ .b) If g ≥ , then gon ( F | K ) ≤ g − .c) If F | K has a prime divisor of degree one, then gon ( F | K ) ≤ g + 1 .d) If in addition g ≥ , then gon ( F | K ) ≤ g . See [16, Lemma 6.6.5] for the proof. In each of the cases of Lemma 2, computation of suitableRiemann-Roch spaces yields explicit elements x ∈ F with [ F : K ( x )] at most the bound given inthe respective case. A FAMILY OF POLYNOMIALS WITH GALOIS GROUP
P SL (2) OVER Q ( T ) 11 Once an exact polynomial equation (over Q or another number field) for a member of a givenfamily of covers - or even for the entire family - has been found, it is necessary to verify the Galoisgroup, especially considering that significant parts of the computations were based on numericalapproximations. There are several easy ways to gain evidence for the Galois group. One of these isthe computation of the monodromy by numerical means; this is a solid tool, although not an exactmethod - and turning it into one requires considerable efforts. However, in all the cases coveredin the following sections, the structure of the Galois group allows for rigorous proofs, which aretherefore given in detail.The following sections will apply the theoretical and computational background to several ex-amples of interest. For each example, the structure will roughly follow the sequence of Sections 2and 3: firstly, a presentation of the properties of the Hurwitz family resp. braid orbit in question,followed by a description of the concrete techniques applied for deformation of covers and retrievingalgebraic dependencies; finally, a presentation of explicit polynomials and verification of their Galoisgroup.
4. A family of polynomials with Galois group
P SL (2) over Q ( t ) We compute a family of coverings with four ramification points, defined over Q , with regularGalois group P SL (2). This yields the (to my knowledge) first explicit polynomials with group P SL (2) over Q ( t ). The group
P SL (2) does not have any rigid triples of rational conjugacy classes, and amongthe genus zero systems of rational class 4-tuples, there is only one with a Hurwitz curve of genuszero. This curve will turn out to be rational in the course of the explicit computations, but thisdoes not seem to be immediately clear by the standard braid orbit criteria (see below). However,if one looks at class 5-tuples, it is possible to obtain P SL (2) as a regular Galois group over Q viapurely theoretical arguments: Proposition 3.
The inner Hurwitz space for the class 5-tuple (2 A, A, B, B, B ) of P SL (2) contains a rational curve over Q , and therefore infinitely many Q -points. A FAMILY OF POLYNOMIALS WITH GALOIS GROUP
P SL (2) OVER Q ( T ) 12 Proof.
This 5-tuple of classes arises as a rational translate of a 4-tuple of classes in
Aut ( P SL (2)).This 4-tuple (of classes (2 A, B, C, A )) has a single braid orbit of length 46; its Hurwitz curve isof genus zero, and the images of the braids in the action on this orbit fulfill an oddness conditionto guarantee the rationality of this genus zero curve.Every Q -point of this rational curve realizes Aut ( P SL (2)) regularly over Q , and as the P SL (2)-fixed field of such a realization is a rational function field (of degree 2 over the base field), one alsoobtains P SL (2).As the explicit computation of such a field extension requires the computation of P SL (2)-coverswith 5 branch points, we content ourselves with a 4-point family in the following. Note however,that the deformation methods of Section 3.1 could be used to obtain members of the above 5-pointfamily from the 4-point one. Let G = P SL (2) in its natural permutation action on 31 points, and denote by 2 A the class ofinvolutions of cycle type (2 . ), by 3 B the class of elements of order 3 with cycle type (3 .
1) in G ,and by 8 A the unique class of elements of order 8 in G (of cycle type (8 . . . SN i ( C ) of class tuples of length 4, of type (2 A, A, B, A ) in G = P SL (5 , G and having product 1, i.e. SN i ( C ) := { ( σ , ..., σ ) ∈ G | σ , σ ∈ A, σ ∈ B, σ ∈ A, h σ , ..., σ i = G, σ · · · σ = 1 } In the notation of Section 2, we have | SN i in ( C ) | = 24. The action of the braid group on SN i in ( C ),as given in Equation (1), is transitive and more precisely yields that there is a family of covers T 7→ C × P C , where C (the C -symmetrized reduced Hurwitz space) is an absolutely irreduciblecurve of genus zero, and for every h ∈ C the corresponding fiber cover is a Galois cover of P C withGalois group P SL (2).Although the usual braid genus criteria yield that the C -symmetrized Hurwitz space for thisfamily is a genus-zero curve, it does not seem clear via standard theoretical considerations (e.g.odd cycle argument for the braid group generators, as in [26, Chapter III. 7.4.]) whether it canalso be defined as a rational curve over Q . In particular, the cycle structure of the braid orbitgenerators acting on the Nielsen class does not yield any places of odd degree. More precisely, theimage of the braid group is imprimitive on the 24 points, with 12 blocks of length 2 (i.e. if F | Q ( t )is the corresponding function field extension, of degree 24, we have an inclusion Q ( t ) ⊂ E ⊂ F ,with [ E : Q ( t )] = 12 and [ F : E ] = 2). As the images in the action on the blocks of the three A FAMILY OF POLYNOMIALS WITH GALOIS GROUP
P SL (2) OVER Q ( T ) 13braids defining the ramification structure of these fields have cycle structure (4 . . . .
2) and(2 . ) respectively, it is clear that E is still a rational function field; however the cycle structureof the latter involution in the action on 24 points is (2 ), so it is possible that a degree-2 placeof E ramifies in F , in which case the rationality of F is not guaranteed. We therefore clarify therationality of this curve by explicit computation.
We start with a degenerate cover with ramification structure (2 A, A, A ), with group P SL (2).We solve the corresponding system of equations for the three-point cover modulo 11, and then liftand retrieve algebraic numbers from the 11-adic expansions. The triple is rigid, but as the conjugacyclass of the element of order 21 is not rational, we obtain a solution over a quadratic number field,namely0 = x · ( x − · ( x − a ) − t · ( x − · x + a ) · ( x − · x + a · x + a ) · ( x − a ) , where ( a , ..., a ) := ( ( −√− , ( −√− , ( √− , ( − √− − , ( −√− σ of class 2 A and σ of class 3 B . One verifies that in all cases, the triple ( σ , σ , ( σ σ ) − )generates an intransitive group isomorphic to P SL (2) × C , with orbits of length 21, 7 and 3. Equa-tions for the genus zero covers induced by this triple on each orbit are easily computed (especiallysince the degree 21 action is imprimitive, so the corresponding equation arises as a composition offunctions of degree 3 and 7) and yield all the information needed for first-order approximations.Now let C ( x ) | C ( t ) be the field extension of rational function fields corresponding to the coverwith four branch points. Via Moebius transformations (in x and in t ) it is possible to assume adefining polynomial f := f ( t, x ) := f ( x ) · ( x − − t · g ( x ) · g ( x ) · x, Closer group theoretic examination yields some evidence for prime divisors of odd degree: namely, the two 3-cyclesof the braid group generator of cycle structure (7 . . ) correspond to degenerate covers with three ramificationpoints, generating two isomorphic, but non-conjugate (in P SL (2)) subgroups. The same holds for the two 2-cyclesof this braid group generator. The explicit computations show that the corresponding prime divisors of ramificationindex 3 and 2 respectively are indeed of degree 1. A FAMILY OF POLYNOMIALS WITH GALOIS GROUP
P SL (2) OVER Q ( T ) 14where deg( f ) = 10, deg( g ) = 2 and deg( g ) = 3 (so we have e.g. assumed the element of order 8to be the inertia group generator over infinity, and the element of order 3 the one over zero). Also,assume that for some λ ∈ C the polynomials f a := f ( a, x ) and f b := f ( b, x ) (where a and b shalldenote the complex zeroes of x + x + λ ) become inseparable in accordance with the elements inthe conjugacy class 2 A .Once we have obtained a complex approximation of such a polynomial f , we now slowly movethe coefficient at x of the above polynomial g to a fixed rational value, and apply Newton iterationto expand the other coefficients with sufficient precision to then retrieve them as algebraic numbers(using the LLL-algorithm). One finds that all the remaining coefficients come to lie in a cubicnumber field. For example, specialization to the rational value − x − x − x −
16, as can be verified with the values in Theorem 4. This already indicates that thereis a rational function field of index 3 in the (genus-zero) function field of the Hurwitz space, whichwould enforce the latter function field to be rational over Q as well. This will be confirmed by thefurther computations. We now choose a prime p such that the above solution, found over a cubic number field, reducesto an F p -point. Any prime such that the defining polynomial of the cubic number field has a singleroot modulo p will do, e.g. p = 11 for our example. Then we apply approach no.3 described inSection 3.2, that is, we lift this point to sufficiently many p -adic solutions (all coalescing modulo p ),in order to obtain algebraic dependencies between the coefficients . These dependencies are all ofgenus zero, and luckily some of them are of very small degree, e.g. if c and c are the coefficientsat x resp. x of the polynomial g , one obtains an equation X i =0 3 X j =0 α ij c i c j = 0of degrees 2 and 3 respectively. As there are a priori (2 + 1) · (3 + 1) = 12 unknown coefficients α ij ,we only need 12 different p -adic liftings to find this dependency as the smallest degree dependencybetween c and c - and maybe a few more to gain evidence that it is not a coincidence. Of course,we find α ij ∈ Q p , but for theoretical reasons we expect them to actually be rational numbers Alternatively, one could just repeat the process of rational specialization and Newton iteration, as above, suffi-ciently often, obtaining cubic minimal polynomials for the other coefficients in each case, and then interpolate.
A FAMILY OF POLYNOMIALS WITH GALOIS GROUP
P SL (2) OVER Q ( T ) 15- and indeed it is easy to retrieve the actual rational numbers from a sufficiently close p -adicapproximation. Next, one easily finds a parameter α for the rational function field defined by thisequation, using Riemann-Roch spaces (cf. Lemma 2).Now, we can express all coefficients as rational functions in α , and obtain the following result: Theorem 4.
Let α, t be algebraically independent transcendentals over Q .Define polynomials f , g , g ∈ Q ( α )[ x ] as follows: f := ( x − α + 1)( α + 4) α − x − α + 1)( α − α − α − α − α + 4) x +8( α + 1)( α − α + 7) x − α + 1) ( α + 12 / α + 3 / α + 106 / α − x + 2 ( α + 1) ( α + 4) α − · ( x + 4 ( α − α + 5 / α + 19 / α + 1) x − α + 42 α + 45 α + 220 α + 4 x −
12 ( α + 1)( α − / α − / α − α − α − α + 4) x + 9 ( α + 1) ( α + 8 / α + 19 / α + 50 / α − x − α + 1) ) ,g := x − x − ( α + 1) ,g := ( x − ( α + 1)( α + 4) α − · ( x + 2 ( α − α + 1) α + 4 x − ( α + 1) ) . Then the polynomial f ( α, t, x ) := f · ( x − − t · g g · x , of degree 31 in x , has Galois group P SL (2) over Q ( α, t ) , with ramification structure (2 . , . , . , . . . with respect to t .Proof. Dedekind reduction, together with the list of primitive groups of degree 31 (as implementede.g. in Magma), shows that
P SL (2) must be a subgroup of the Galois group. It therefore sufficesto exclude the possibilities A and S .Multiplying t appropriately, we can assume the covers to be ramified in t = 0 , t = ∞ and thezeroes of t + t + λ , with some parameter λ . Interpolating through sufficiently many values of α onefinds the degree-24 rational function λ = h ( α ) h ( α ) parameterizing the Hurwitz curve. As e.g. α = 0and α = 1 / λ , we set t = C · ( f · ( x − g · g · x )(0 , s ) (evaluating x to a parameter s of a rational function field, as well as α to 0, and multiplying with a suitable constant C to obtainthe above condition on the branch points). Then one can check that over Q ( s ), the polynomial f (1 / , C · t, x ) (again for a suitable constant C to obtain the branch point conditions) splits intotwo factors of degrees 15 and 16. This means that for this particular point of the family, there is anindex-31 subgroup of the Galois group that acts intransitively on the roots. As P SL (2) has sucha subgroup and A and S don’t, the Galois group for this particular specialization is P SL (2).This specialization corresponds to an unramified point on the (irreducible) Hurwitz space, thereforethe entire family must belong to the same Hurwitz space and therefore have Galois group P SL (2)over Q ( α, t ).We can now specialize α to any value that does not let two or more ramification points coalesce,to obtain polynomials with nice coefficients with group P SL (2) over Q ( t ). E.g. α Corollary 5.
The polynomial ˜ f ( t, x ) := ( x − x − x − x − x − ( x + 4 x − x + 56 x + 53 x − ( x − − t ( x − x − ( x − x − ( x + 2) x ∈ Q ( t )[ x ] defines a regular extension of Q ( t ) with Galois group P SL (2) . POLYNOMIALS WITH GALOIS GROUP
P SL (4) ≤ G ≤ P Γ L (4) OVER Q ( T ) 16In fact it can be seen from λ = h ( α ) h ( α ) (as in the proof above) that the only specialized rationalvalues for α that lead to degenerate covers (with two branch points coalescing) are α
7→ − α
7→ − α Remark:
The above proof essentially uses the fact that
P SL (2) has two non-conjugate actions on 31 pointsinducing the same permutation character. This can of course be applied to other linear groups,and has e.g. been used in [23] to verify P SL (11) (and others) as the Galois group of a family ofpolynomials. Cf. also the Galois group verifications in the following sections.
5. Polynomials with Galois group
P SL (4) ≤ G ≤ P Γ L (4) over Q ( t ) Previously, there have not been any explicit polynomials f ( t, X ) ∈ Q ( t )[ X ] with regular Galoisgroup P Γ L (4)(= P SL (4) .S ) , P GL (4)(= P SL (4) .
3) or
P SL (4). Malle gave a polynomial for P SL (4) . P SL (4) by the field automorphism) in [24, Theorem 3], but this doesnot yield a P SL (4)-polynomial, as the P SL (4)-fixed field does not have genus 0 (see however [32,p.2] for a way to obtain from Malle’s polynomial a P SL (4)-polynomial over Q (not Q ( t )).Theoretical arguments for all P SL (4) ≤ G ≤ P Γ L (4) to be a regular Galois group over Q ( t )have however been known for a long time (cf. [26], Example 4.2. in Chapter IV.4). We find polynomials for all groups
P SL (4) ≤ G ≤ P Γ L (4) by computing the Hurwitz spaceof a family of covers with Galois group P Γ L (4), ramified over four places with ramification struc-ture (2 . , . , . , .
1) with regard to the natural degree 21 permutation representation of P Γ L (4). The length of the corresponding Nielsen class is 20, and the C -symmetrized inner Hur-witz curve is a rational curve of genus zero. Therefore this family leads to many polynomials withregular Galois group P Γ L (4) over Q . The fixed field of P SL (4) in such an extension is still ofgenus zero, as can be seen by the coset action of the above class four-tuple on P Γ L (4) /P SL (4).However, even the fixed field of P GL (4) cannot be guaranteed to be a rational function field bytheoretical means (it is a genus-zero degree-2 extension of the function field Q ( t ), ramified in twoplaces, which are possibly algebraically conjugate, in which case the extension field need not be POLYNOMIALS WITH GALOIS GROUP
P SL (4) ≤ G ≤ P Γ L (4) OVER Q ( T ) 17rational). The above fixed field would automatically be rational for any rational point on the un-symmetrized Hurwitz curve - i.e. for a regular P Γ L (4)-extension with all branch points rational -but this curve is not of genus zero anymore.We therefore verify by explicit computation that the fixed field of P SL (4) is indeed a rationalfunction field for suitable choices of parameters; this yields explicit polynomials with regular Galoisgroups P GL (4) and P SL (4) as well. P Γ L (4)The deformation and algebraization process for our family is analogous to the one in Section 4(note that the Hurwitz curves are rational in both cases). It should therefore suffice to present theresulting polynomial. We only note briefly that a good permutation triple to start the deformationprocess from is the triple with cycle structures (2 . , . . , . P SL (4) .
2. A polynomial for this triple is easily found modulo a small prime and then liftedto a polynomial defined over a number field - in this case Q ( √− Theorem 6.
The polynomial f := ( x + ( α − x − ( α + 20) x + 5 α ) ( x + 1) x − t (( α − α + 45) x + 18 ( α − α + 85 α − α + 1476) x + 12 ( α − α + 53 α − α + 360) x + 14 α ( α − α + 77 α − x + 12 α ( α − α + 65) x + 18 α ( α − ) · ( α ( α + 3) x + (4 α − α + 47 α + 192) x + 2(2 α − α + 127 α − α + 880) x +2(2 α − α + 347 α − α + 3000) x + ( − α + 405 α − α + 9000) x + 125( α − α + 40)) ∈ Q ( α, t )[ x ] has regular Galois group P Γ L (4) over Q ( α, t ) , with ramification structure (2 . , . , . , . with regard to t .Proof. Specializing in appropriate finite fields, one sees that the Galois group of f is either P Γ L (4)or S . Now P Γ L (4) has two non-conjugate subgroups U and V of index 21. If one considers theaction of P Γ L (4) on the right cosets of U , then V is intransitive with orbits of length 5 and 16.The images of the desired inertia subgroup generators σ , ..., σ in the action on the cosets of V arestill of the same cycle type, and therefore belong to the same family of covers, but not to the samecover.A suitable linear transformation in t assures that the function field extension Q ( α )( x ) | Q ( α )( t )is ramified over t t
7→ ∞ and t
7→ { zeroes of t + t + µ ( α ) } for some rational function µ ( α ) ∈ Q ( α ). This choice of ramification yields a good model for the C -symmetrized Hurwitz curve.One then notes that the specializations α
10 and α
13 lead to the same ramification locus. If f ( t, x ) = p ( x ) − tq ( x ) and f ( t, x ) = p ( x ) − tq ( x ) are the corresponding polynomials, thepolynomial p ( x ) · q ( y ) − q ( x ) · p ( y ) decomposes in Q [ x, y ] into factors of degree 5 and 16.This means that there is an index-21 subgroup in the Galois group of f ( t, x ) acting intransitivelywith orbits of length 5 and 16. Therefore f must have Galois group P Γ L (4), and as our Hurwitzspace is connected, the same must hold for the two-parameter polynomial f . POLYNOMIALS WITH GALOIS GROUP
P SL (4) ≤ G ≤ P Γ L (4) OVER Q ( T ) 18 P Γ L (4)As noted above, the fixed field of P GL (4) in the Galois closure of f is of genus zero. It is givenas Q ( α )( X, Y ), where p α ( X, Y ) := X + 3( Y − ( α − α + 90)( α − α + 40)) = 0. Althoughthe conic given by p α ( X, Y ) = 0 does not split generically - i.e. it does not have any Q ( α )-rationalpoints, there are many values α ∈ Q for which the specialized curve given by p α ( X, Y ) = 0 hasnon-singular points, which means that the residue field Q ( X, Y )[ α ] / ( α − α ) is a rational functionfield for these values α α , i.e. it can be parametrized as Q ( s ). One such example is α = 10. Inthis case, parametrizing t as a rational function in s yields the following polynomial, with regularGalois group P GL (4) over Q : g := ( s + 3)( x − x + 50) ( x + 1) x − · · · (111151 s + 389344 s − · (85 x +1582 x +5140 x − x +7250 x +625 / (130 x +3162 x +20580 x +13700 x − x +11250) ∈ Q ( s )[ x ]As the fixed field of P SL (4) is a degree 3 genus zero extension of the fixed field of P GL (4), it is arational function field whenever the latter field is. Parameterizing it for our specialization α = 10leads to the following polynomial with regular Galois group P SL (4): h := ( y − y + 1) ( x − x + 50) ( x + 1) x − ( 4751689 y − y − y + 27261652396985753125 y − y − y + 4751689) · (85 x +1582 x +5140 x − x +7250 x +625 / (130 x +3162 x +20580 x +13700 x − x +11250) ∈ Q ( y )[ x ] P SL (4) ≤ G ≤ P Γ L (4)The family computed above does not yield any totally real Galois extensions with the abovegroups, as can be checked easily by observing the action of complex conjugation on the class tuplesin the Nielsen class. This conjugation is never given by the identity element of P Γ L (4), whichwould however be necessary to obtain a totally real specialization.On the other hand, the family used in [26], Example 4.2 in Chapter IV.4 to obtain P SL (4) ≤ G ≤ P Γ L (4) as regular Galois groups by theoretical means does lead to such specializations. Icomputed this family in an earlier version of this paper (cf. [20, Chapter 7]); it also has a rational POLYNOMIALS WITH GALOIS GROUP
P SL (4) ≤ G ≤ P Γ L (4) OVER Q ( T ) 19Hurwitz curve, but the corresponding polynomials turned out to have rather large coefficients,therefore I will only give a single polynomial for P Γ L (4). Polynomials for proper normal subgroupscan be obtained from this in the usual way. Theorem 7.
The polynomial f ( t, x ) := ( x + 18453672844570518827351464949935671 x − x + 280994743496912162168745209999690339071182011991406356852970921579 x + 215030295468310342214055204414793418468315707162781145761406356852970921579 x − x + 29340262301998944183599512791769174814051478361133335734219020448492804672281903557879 x + 32207444140741785418416092872399487301042274723030519123457190116033359797764672281903557879 ) · ( x + 662967398108898410049 x − x + 197524237576626624310401860686399324846059576029860583050014703215594045637427773689649 x − x − x + 4024986270625461332271391750272716366360754561529477022417825216953307017955915232443247262613504564610511697807282801 x + 430514906251822635197320013134955148658731797346020041700003042831393164490244634750377158001 ) − t ( x + 29134352928410049 ) ( x − ( x − ( x − x has regular Galois group P Γ L (4) over Q ( t ) . The ramification structure with regard to t is of type (2 . , . , . , . . . . Furthermore, let a be the unique ramification point of f inside thereal interval ( −∞ , , i.e. a ≈ − . · . Then for t ∈ ( a, , the specialized polynomial f ( t , x ) is totally real. TOTALLY REAL EXTENSIONS WITH GROUP
P GL (11) 20
6. Totally real extensions with group
P GL (11) In this section and the following ones, we will focus on totally real extensions. In particular,we compute explicit polynomials for totally real Galois extensions over Q , with Galois groups P GL (11) = P SL (11) . P SL (3), M and Aut ( M ) = M .
2. The first two of these groups arethe smallest (with respect to minimal faithful permutation degree) that have not been previouslyrealized as the Galois group of a totally real extension of Q , this means that explicit totally realpolynomials are now known for every transitive permutation group of degree at most 13 (cf. [19]).By now, some totally real specializations of the P GL (11)- and P SL (3)-polynomials computedbelow have been inserted in the database [19].Note that totally real extensions can only be obtained via families with four or more branchpoints, cf. [26], Chapter I, Example 10.2. The problem for the group P GL (11) is that, on the onehand, to obtain totally real fibers (i.e. a complex conjugation acting as the identity) one needs tocompute polynomials with at least four branch points. On the other hand P GL (11) in its naturalaction has no generating genus zero tuples of length r ≥
4. There are however genus zero tuples inthe imprimitive action on 22 points, which stems from the exceptional action of
P SL (11) on 11points (this degree-11 action was also used by Malle to compute totally real P SL (11)-polynomialsin [23, Section 9]). Below are explicit computations for two such class tuples. (2 A, B, B, A ) Firstly, let C = (2 A, B, B, A ) the quadruple of classes of P GL (11), where 3 A is the uniqueclass of elements of order 3, 2 A is the class of involutions inside P SL (11), and 2 B the class ofinvolutions outside P SL (11). This is a genus zero tuple in the imprimitive action on 22 points, sofor a degree-22 cover of P ( C ) with this ramification type, we get the following inclusion of functionfields: C ( t ) ⊆ C ( s ) ⊆ C ( x ), where exactly two places of C ( t ) ramify in C ( s ) (namely the ones withinertia group generator not contained in P SL (11)), and exactly four places of C ( s ) ramify in C ( x )(namely two places lying over the ramified place of C ( t ) with inertia group generator in 2 A , andtwo lying over the place of C ( t ) with inertia group generator 3 A ).The essential task is therefore to compute the extension C ( x ) | C ( s ) , i.e. to compute polynomialswith P SL (11)-monodromy, defined over Q if possible, and ramification type (2 A, A, A, A ). Thestraight inner Nielsen class of these tuples in P SL (11) is of length | SN i in | = 54, with transitive TOTALLY REAL EXTENSIONS WITH GROUP
P GL (11) 21braid group action and symmetrized braid orbit genus g = 1. Via Moebius transformations, wetherefore assume that the two places of C ( s ) with inertia group generator of order 3 are s s
7→ ∞ , and also fix the sum of the other two branch points. As the cycle structure of an element σ in the class 3 A of P SL (11) in the action on 11 points is (3 . ), and one of the 3-cycles remainsfixed under conjugation with N P SL (11) ( h σ i ) (and therefore under the action of the decompositionsubgroup), one can assume w.l.o.g. for a model over Q that the place x s x
7→ ∞ and s
7→ ∞ . That is, we may w.l.o.g. look forpolynomial equations x · f ( x ) · f ( x ) − s · g ( x ) · g ( x ) = 0, with quadratic polynomials f i , g i . Due to the relatively small degree, one can immediately search for a mod- p reduced polynomialwith the above restrictions on places and the correct ramification, instead of starting with a 3-pointcover and going through the deformation process. There is a solution with the correct Galois groupover F .Now lift this solution to many approximate Q -solutions, with the set of zeroes of s · ( s + 4 s + λ )as the finite ramification locus (for many different values of λ ). Interpolation then yields an algebraicdependency between the coefficients at x of the above polynomials g and g , namely:(88 / β − / β + 32 / α + (178 / β − / β + 446 / β − / α +(287 / β − / β + 2051 / β − / β + 295 / α +(59 / β − / β + 773 / β − / β + 435 / β − / α +10 / β − / β + 21 / β − / β + 491 / β − / β + 1 = 0(with α the coefficient of g and β the one of g ). Computation with Magma confirms that thisdefines an elliptic curve of rank 1 (more precisely, this curve can be defined by the cubic equation Y = X − X − α and β , therefore this curve is already a model ofthe reduced Hurwitz curve of the P SL (11)-family. So there are infinitely many equivalence classesof covers defined over Q with this monodromy. Additional symmetrization of the branch points 3 and 4 does not decrease this genus.
TOTALLY REAL EXTENSIONS WITH GROUP
P GL (11) 22However, as we are interested in totally real polynomials, we need to choose a point on thecurve in such a way, that complex conjugation on a fiber of the corresponding P SL (11)-cover istrivial in at least one segment of the punctured projective line. Monodromy computations showthat α = − and β = yields such a point. This leads to the polynomial f ( s, x ) := x ( x + x − ( x − x + 63181016) − s ( x − x + 5671131350) ( x + 4155 x − , (2)where specializations of s in the real interval [ − . .., − . .. ] (between the two algebraicallyconjugate branch points) lead to totally real fibers.Now all that is left is to parameterize the above extension C ( s ) | C ( t ) over Q to fit the positionsof the branch points. This leads to the following: Theorem 8.
Let f ( x ) := x ( x + x − ( x − x + 63181016) ,f ( x ) := ( x − x + 5671131350) ( x + 4155 x − , and F ( t, x ) := f ( x ) + 2728079147653721954955473000 f ( x ) f ( x ) + 7663094829906251985274409206528 f ( x ) − tf ( x ) f ( x ) ∈ Q ( t )[ x ] . Then F has regular Galois group P GL (11) over Q ( t ) and possesses totally real specializations forall t t > (i.e. t larger than the largest finite branch point). The branch cycle structurewith respect to t is of type (2 . , , , . ) .Proof. F is gained from the polynomial f in (2) by setting t := ( s + 2728079147653721954955473000 s + 7663094829906251985274409206528) /s. We therefore first prove that f has Galois group P SL (11).As in Section 5, we compute an explicit algebraic dependency for the natural (degree 54) coverof the reduced Hurwitz space over P . We use this to find a second cover with the same ramificationlocus as the one given by f , and then make use of the fact that P SL (11) has two non-conjugatesubgroups of index 11. Set˜ s = − ( 295726 ) · s · ( s + s + 693 / · ( s + 1107 / · s − / s + 297 / · s − / · ( s + 46 / · s + 12474 / . Then f (˜ s, x ) splits over Q ( s ) into polynomials of degree 5 and 6. This shows that Gal ( f | Q ( s )) hasan intransitive index-11 subgroup, and so it cannot be equal to A or S . Dedekind reductionthen leaves only P SL (11). So f has Galois group P SL (11) over Q ( s ), and regularity is obvious.Therefore Gal ( F | Q ( t )) is a transitive subgroup of the wreath product P SL (11) ≀ C < S . Now one TOTALLY REAL EXTENSIONS WITH GROUP
P GL (11) 23can check immediately that the only transitive subgroup of this wreath product with a generating 4-tuple (with product 1) of the necessary cycle structure is P GL (11). So P GL (11) is the geometricGalois group of F , and regularity follows because P GL (11) is self-normalizing in S .Finally, the assertion about totally real specializations is easy to verify. (2 A, A, B, A ) We consider another family, namely (in analogy to the above notation) the one associatedto the class quadruple (2 A, A, B, A ) in P GL (11). Again, looking at the imprimitive action of P GL (11) on 22 points, this monodromy leads to function fields C ( t ) ⊆ C ( s ) ⊆ C ( x ). This time, the P SL (11)-part C ( x ) | C ( s ) is ramified over 5 points, with monodromy of type (2 A, A, A, A, A ).We therefore look for points on a reduced Hurwitz space of dimension 2. However, we do not needto parameterize the whole surface.Suitable choice of the branch points in C ( t ) and C ( s ) leads to a model for a two-parameterpolynomial, corresponding to a curve on the Hurwitz space. Firstly, we can map the branch pointsof C ( t ) to 0, ∞ and − ± α , with α ∈ Q (for a rational model) and only the places at zero andinfinity ramifying in C ( s ). Therefore, by setting t = s , we may assume that the finite ramificationlocus of s in C ( x ) is ±√− − α , ±√− α , and therefore the set of zeroes of the polynomial s + 2 s + (1 − α ) =: s + 2 s + λ .We can use the braid criteria exhibited in [9] to confirm the existence of a cover C → P ,where C is a curve of genus 1, parameterizing the polynomials with the above monodromy andrestrictions on branch points. More precisely, our restrictions on the branch points lead to thesame braids ( R := β β and R := β β β ) as curve no. 14 on p. 49 in [9]; the group generatedby these braids acts intransitively on the inner Nielsen class of P SL (11)-generating tuples oftype (2 A, A, A, A, A ), with an isolated orbit of length 48 and corresponding braid orbit genus1. (Alternatively, observe that the 4-tuple in P GL (11) with which we started to obtain therestrictions on the branch points has a Hurwitz curve of genus 1.) As a starting point for the computations, we used a polynomial with 4 branch points and
P SL (11) monodromy, computed by Malle in [23]. Develop this into a cover with 5 branch points(as done in the previous examples), and observe that the normalizer of an involution in P SL (11)fixes one of the 2-cycles, therefore we can assume a polynomial equation f ( x ) − s · g ( x ) · g ( x ) = 0, TOTALLY REAL EXTENSIONS WITH GROUP
P GL (11) 24with deg ( f ) = 11 and deg ( g i ) = 3 (i.e. the infinite place of C ( x ) lies over the infinite place of C ( s ),with ramification index 2).Specializing the coefficients of g and g at x to sufficiently many rational values again allowedan interpolation polynomial (of degree 4 in both variables), and Magma computation again yieldsthat this polynomial defines an elliptic curve of rank 1.Now the procedure is the same as for the previous family: find a point on this curve that allowsfor a totally real fiber cover (one such point yields the polynomial g ( s, x ) := ( x − x + 9 x + 11 x − x − x − / · ( x + 14 x − / x − x + 1 / x + 5 / − s ( x + x − / ( x + 4 x + 5 x + 18 / P SL (11)), and compose the resulting parameterization of s as a rational functionin x with t = s .The Galois group can of course be verified just like in Theorem 8. In this case, we also computeda degree-12 polynomial defining the stem field of a stabilizer in P GL (11) in its natural action on12 points. This is the polynomial ˜ g in Theorem 9 below. It was found in the following way: Let E be the splitting field of the above polynomial g over Q ( s ). A primitive element of a subfield of E of degree 12 over Q ( s ) (corresponding to the stabilizer in P SL (11) in its action on 12 points)can be computed with Magma. From this, one obtains a primitive element of the correspondingdegree-12 extension of Q ( s ) as well. By the Riemann-Hurwitz genus formula, this field is of genus2. Therefore its gonality is 2. Via computation of Riemann-Roch spaces a rational subfield ofindex 2 can explicitly be parameterized. A few linear transformations then yielded the followingpolynomial: Theorem 9.
The polynomial ˜ g ( t, x ) = (( x + x + 14 x + 122 ) ( t + 1249) − x + 57 x −
144 )( x − x − x − x + 36143 x − x − x − x + 5715138424 x − t − · · ·
114 ( x + 2 x + 321550 x − x − x + 4015324 ) ( x + 632693 x − has regular Galois group P GL (11) over Q ( t ) . The branch cycle structure with respect to t isof type (2 , , . , ) . Furthermore, if a is the unique branch point of ˜ g inside (0 , + ∞ ) , i.e. a ≈ . the positive root of t + 1249 t − / , then for t ∈ (0 , a ) , the specializedpolynomial ˜ g ( t , x ) is totally real. TOTALLY REAL EXTENSIONS WITH GROUP G = P SL (3) 25
7. Totally real extensions with group G = P SL (3) Computing totally real
P SL (3)-extensions might be possible via covers with four branch points;however, there are no genus zero 4-tuples with a Hurwitz curve of genus zero in P SL (3). Wetherefore solve the problem via a family of covers with five branch points, with branch cycle structure(2 . , . , . , . , . ) in the natural permutation representation of P SL (3). The reasonis that this family can be seen to give rise to totally real P SL (3)-extensions via purely theoreticalcriteria: Proposition 10. In P SL (3) (in its natural degree 13 action), let A be the class of involutionsof cycle type . and A be the class of elements of cycle type . . Then the inner Hurwitzspace of C := (2 A, A, A, A, A ) contains a rational genus zero curve over Q , and thereforeinfinitely many Q -points. Furthermore, among these Q points, there are some that lead to totallyreal P SL (3) -polynomials.Proof. The group generated by the braids B := β β β and B := β β acts intransitively on the120 P SL (3)-generating 5-tuples of the straight inner Nielsen class SN i in ((2 A, A, A, A, A )).This braiding action corresponds to curve no. (26) given on p.51 in Dettweiler’s list of curves onHurwitz spaces in [9]. The orbits under this action are of lengths 12, 48 and 60; and the cyclestructure of the braids in the action on the orbit of length 12 yields a (rational) genus zero curveon the Hurwitz space.Alternatively, observe that the 4-tuple of classes in Aut ( P SL (3)) (as an imprimitive permuta-tion group on 26 points) with cycle structures (2 . , , . , . ) has braid orbit genus g = 0.Our P SL (3)-5-tuple becomes a rational translate of this 4-tuple in a natural way, via ascendingto the P SL (3)-fixed field. Therefore every rational point on the genus zero Hurwitz curve for the4-tuple also yields a regular realization of P SL (3) with the desired monodromy.The statement about totally real polynomials follows from group theoretic considerations. Oneonly needs to find an element of our braid orbit where the identity element of P SL (3) acts ascomplex conjugation on the branch cycles, as described in [26, Thm. I.10.3]. This yields the existenceof P SL (3)-covers with totally real fibers, and as rational points are dense around real points onour g = 0-Hurwitz curve, there are also such covers defined over Q . As a starting point for the computations, we use a 4-point cover with group
P SL (3), withbranch cycle structure (2 A, A, A, A ), as computed by Malle in [23]. From this, the usual defor-mation process of Section 3.1 leads to a 5-point cover with the above cycle structure, after writingthe element of order 4 as a product of two involutions in P SL (3).Once this is achieved, Proposition 10 yields a recipe to compute a two-parameter family of P SL (3)-polynomials - parametrized by a rational curve on the Hurwitz space - and specializeappropriately to obtain totally real extensions. This has been carried out in [20, Chapter 8.2]. The TOTALLY REAL EXTENSIONS WITH GROUP G = P SL (3) 26computations are analogous to the ones that have been performed several times by now. However,it turned out that the same ramification type also yields a three-parameter family of covers definedover Q - something that did not seem obvious from the theoretical arguments. We therefore describethe computation leading to this stronger result. Explicit computations show that the reduced Hurwitz space H , consisting of equivalence classesof covers with partially ordered branch point set ( { zeroes of t + t + at + b } , , ∞ ) (with parameters a, b ) and monodromy as above, does not only contain rational curves, but is in fact a rationalsurface. Its function field is therefore of the form Q ( α, β ) with independent transcendentals α , β .In other words, there is a three-parameter family f ( α, β, t, x ) of P SL (3)-polynomials over Q ( t ),with branch point restrictions as above. This family was found, beginning with any member of thetwo-parameter family in [20, Chapter 8.2], by once again applying the techniques of Section 3.2.Lifting an inital mod- p solution to many different polynomials with the above restrictrion on branchpoints yielded an algebraic equation between three suitable coefficients, say α, γ and δ . Luckily,the curve given by this equation over the constant field Q ( α ), was of genus zero, and even rational.Riemann-Roch space computations therefore yield its parameter β - as a rational function in α, γ and δ . Finally, algebraic dependencies between α , β and any of the remaining coefficients of themodel lead to the following nice result: Theorem 11.
The polynomial f ( α, β, t, x ) := f · f · x − t · g · g ∈ Q ( α, β )( t )[ X ] , with f := x + βx + ( β − x − αβ + 49 αβ − α,f := x + αβ − αβ + 12 α − β − β − x + αβ − αβ + 12 α − β − β − x − ,g := x + αx + 13 αβx + 19 αβ − α,g := αx + 4 αβ − α + 93 x + 4 αβ − αβ + 9 α + 9 β − x − α. has regular Galois group P SL (3) over Q ( α, β ) . Suitable specializations for α, β and t yield totallyreal P SL (3) -extensions.Proof. The Galois group can again be verified using the two non-conjugate index-13 subgroups of
P SL (3). As for totally real extensions, it would be somewhat complicated to classify all possibilitiesfor specializations of α, β and t . We therefore content ourselves with the special case α
7→ − β
7→ −
6. In this case, all choices t t with t between the two smallest real branch points, i.e. t ∈ ( − . .., − . .. ), yield totally real specializations. TOTALLY REAL EXTENSIONS WITH GROUPS M AND
AU T ( M ) 27The above family leads to P SL (3)-polynomials with various other ramification types as well.One particularly interesting observation is that f ( α, β, t, x ) (as a polynomial in x ) also definesa genus zero extension with respect to α (not just with respect to t !), although not in rationalparameterization. The branch cycle structure with respect to α consists of six involutions (all ofcycle structure (2 . )). Remark:
The next open cases with regard to totally real Galois extensions occur for the permutation degree n = 14: there are no explicitly known totally real Galois extensions of Q with Galois group P SL (13)or P GL (13). For these groups, the genus zero approach will no longer work. This is obvious for P GL (13), as this group does not possess any generating genus zero tuples of length ≥
4. For
P SL (13), there is just one rational genus zero 4-tuple (of cycle type (2 A, A, A, A )), with aHurwitz curve of genus g = 1. One might therefore hope for an elliptic curve of rank ≥
1, as in theabove
P GL (11)-cases. However, explicit computation showed that this is an elliptic curve of rankzero (and more precisely, can be defined by y = x − x + 136 x −
8. Totally real extensions with groups M and Aut ( M ) The automorphism group
Aut ( M ) = M . M has rational Hurwitzcurves for genus zero 4-tuples, which however do not give rise to totally real specializations. Wetherefore computed the Hurwitz space for the family of cycles structures (2 . , . , , . . ).The braid orbit is of length 30, and the Hurwitz curve of genus 1. Once again, the elliptic curveturns out to be of rank one, and does indeed provide rational points belonging to totally real fibers.We only give one example: Theorem 12.
The polynomial f ( t, x ) := 9180125( x + 77 x − ( x + 2816) (2439 x − x + 10912) − t (367205 x +59565800 x − x − x − x +556611262821376 x +1682125644320768 x − x + 14606802351030272) (405 x +52290 x +5828131 / x − / x +21649071627 / x − x +3867113368) has regular Galois group Aut ( M ) over Q ( t ) , with ramification structure (2 . , . , , . . ) with regard to t . For all t > , the specialized polynomial f ( t , x ) is totally real.Furthermore, after setting t := t ( s ) := (11 s + 1) / ( − · · · · · s + 1)) , the polynomial g ( s, x ) := f ( t ( s ) , x ) has regular Galois group M over Q ( s ) and yields totally real specializationsfor all s s ∈ Q with | s | < . . CONCLUDING REMARKS AND APPLICATIONS Proof.
The assertions about totally real specializations can be easily checked; also after proving thefirst assertion, one simply computes the discriminant to confirm that the Galois group of g mustbe Aut ( M ) ∩ A = M .So we are left with showing that Gal ( f | Q ( t )) ∼ = Aut ( M ). By Dedekind reduction, one quicklysees that the only candidates are Aut ( M ) and S . To exclude the latter, one may use the factthat Aut ( M ) has an index-77 subgroup acting intransitively with orbits of degrees 6 and 16 (thestabilizer of a block of the (3 , , t
7→ ∞ form such a block, expand the six simple poles of t = t ( x )as series in t . The symmetric functions in these six elements then generate the fixed field F ofthe Steiner block stabilizer. Sufficiently precise series yield an algebraic dependency describing F ,and as the Riemann-Hurwitz formula shows that this field is of genus 2, suitable Riemann-Rochspace computations even yield an equation in two variables of degrees 2 and 3 for F . Now express t through these two variables; so far, everything has been based on approximations, but now verifythat the polynomial f ( t, x ) decomposes over (the genus 2 field) F into factors of degree 6 and 16.Riemann-Hurwitz shows that the fixed field of a 6-set stabilizer in S would have much highergenus; this proves the assertion. (Unfortunately, the occurring equations are too large to fit intothis paper; however, note that once again this method of proof is rigorous and does not rely onnumerical approximation as a monodromy computation would.)
9. Concluding remarks and applications
All the Hurwitz spaces under consideration in the previous sections turned out to possess in-finitely many rational points. In particular, Theorems 8, 9 and 12 provide a single polynomial withreal fibers, corresponding to a rational point on the respective Hurwitz curve. As mentioned above,there are actually infinitely many rational points, as the curves are elliptic of rank rk >
0. Asfor any non-singular cubic curve E , defined over Q , with E ( Q ) infinite, Q -points lie dense (in thetopology of P R ) around any given rational point, one obtains as an immediate corollary that theHurwitz curves of all these families contain infinitely many points with real fibers. This is becausethe property to possess real fibers is purely group theoretic and therefore locally invariant in theHurwitz space.An obvious application of the parametric families is the search for number fields with prescribedGalois group (and possibly prescribed signature) and small discriminant, cf. the Kl¨uners-Malledatabase [19]. We only give two examples. The first is a totally real P GL (11)-polynomial withroot discriminant of small absolute value. Lemma 13.
The polynomial g ( x ) := x − x − x − x + 9768 x + 18480 x − x − x +352880 x + 1664960 x + 455488 x − x + 217152 has Galois group P GL (11) over Q and splitting field contained in R . The discriminant of a rootfield is equal to · · · ≈ . EFERENCES g in Theorem 9 by specializing t /
10 andthen applying Magma’s method
OptimizedRepresentation .Also, the
P SL (3)-family from Theorem 11 has many specializations with “small” discriminantin the sense that very few primes ramify. We conclude by giving a P SL (3)-number field ramifiedover one prime only. Lemma 14.
The polynomial f ( x ) := x − x − x − x + 8156061 x + 770464590 x +11462215447 x − x − x + 18705429494567 x +29408002566579439 x + 237585722590314749 x − x − has Galois group P SL (3) , and only the prime p = 83420911386433 ramifies in its splitting field.In fact, a root field has discriminant p . This polynomial is obtained from Theorem 11, via α := 148 / β := 1 and t := −
1, and againMagma’s
OptimizedRepresentation . Acknowledgement
I would like to thank Peter M¨uller for introducing me into many of the subjects of this workas well as carefully reading earlier versions, J¨urgen Kl¨uners for informing me about some opencases of totally real Galois extensions, and the anonymous referee for many helpful suggestions forimprovements.
References [1] A.O.L. Atkin, H.P.F. Swinnerton-Dyer,
Modular forms on noncongruence subgroups . Combina-torics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Amer.Math. Soc., Providence, R.I. (1971), 1–25.[2] P. Bailey, M. Fried,
Hurwitz monodromy, spin separation and higher levels of a modular tower .Proc. Sympos. Pure Math 70, Amer. Math. Soc. (2002), 79–220.[3] W. Bosma, J. Cannon, C. Playoust,
The Magma algebra system I: The user language . J. Symb.Comput. 24 (1997), 235–265.[4] J.-M. Couveignes,
Tools for the computation of families of coverings . Aspects of Galois theory(Gainesville, FL, 1996), London Math. Soc. Lecture Note Ser., 256, Cambridge Univ. Press,Cambridge (1999), 38–65.
EFERENCES
Boundary of Hurwitz spaces and explicit patching . J. Symb. Comput. 30(2000), 739–759.[6] J.-M. Couveignes, L. Granboulan,
Dessins from a geometric point of view . Leila Schneps (editor),The theory of Grothendieck’s dessins d’enfants, Cambridge University Press (1994), 79–113.[7] P. Debes, M. Fried,
Rigidity and real residue class fields . Acta Arithmetica 56.4 (1990), 291–323.[8] P. Debes, M. Fried,
Non-rigid constructions in Galois theory . Pacific J. Math. 163, No. 1 (1994),81–122.[9] M. Dettweiler,
Kurven auf Hurwitzr¨aumen und ihre Anwendungen in der Galoistheorie . PhDThesis, Erlangen (1999).[10] M. Fried, H. V¨olklein,
The inverse Galois problem and rational points on moduli spaces . Math.Ann. 290 (1991), no. 4, 771–800.[11] L. Gerritzen, F. Herrlich, M. van der Put,
Stable n -pointed trees of projective lines . Indag.math. 50 (1988), 131–163.[12] L. Granboulan, Construction d’une extension reguli`ere de Q ( T ) de groupe de Galois M .Experiment. Math. 5 (1996), no. 1, 3–14.[13] E. Hallouin, Study and computation of a Hurwitz space and totally real
P SL ( F ) -extensionsof Q . J. Alg. 321 (2009), 558–566.[14] E. Hallouin, E. Riboulet-Deyris, Computation of some moduli spaces of covers and explicit S n and A n regular Q ( t ) -extensions with totally real fibers . Pacific Journal of Math. 211, No. 1(2003), 81–99.[15] A. Hurwitz, ¨Uber tern¨are diophantische Gleichungen dritten Grades . Vierteljahrschr. d. Naturf.Ges. in Z¨urich 62 (1917), 207–229.[16] M. Jarden, Algebraic Patching . Springer Monographs in Mathematics, Berlin-Heidelberg(2011).[17] M. Klug, M. Musty, S. Schiavone, J. Voight,
Numerical calculation of three-point branchedcovers of the projective line . Preprint (2013), available at http://arxiv.org/abs/1311.2081 . EFERENCES
Explicit Galois realization of transitive groups of degree up to 15 . J.Symb. Comput. 30 (2000), 675–716.[19] J. Kl¨uners, G. Malle,
A database for field extensions of the rationals . LMS Journal of Compu-tation and Mathematics 4 (2001), 182–196. Database at http://galoisdb.math.upb.de/ [20] J. K¨onig,
The inverse Galois problem and explicit computation of families of covers of P C with prescribed ramification . Disseration, W¨urzburg (2014).[21] A.K. Lenstra, H.W. Lenstra, L. Lovasz, Factoring polynomials with rational coefficients . Math.Ann. 261 (1982), 513–534.[22] K. Magaard, S. Shpectorov, H. V¨olklein,
A GAP package for braid orbit computation andapplications . Experimental Math. Vol. 12 (2003), No. 4, 385–393.[23] G. Malle,
Multi-parameter polynomials with given Galois group . J. Symb. Comput. 21 (2000),1–15.[24] G. Malle,
Polynomials with Galois groups
Aut( M ) , M , and PSL ( F ) · over Q . Math.Comp. 51 (1988), 761–768.[25] G. Malle, Polynomials for primitive nonsolvable permutation groups of degree d ≤
15. J. Symb.Comput. 4 (1987), 83–92.[26] G. Malle, B.H. Matzat,
Inverse Galois Theory . Springer Monographs in Mathematics, Berlin-Heidelberg (1999).[27] P. M¨uller,
A one-parameter family of polynomials with Galois group M over Q ( t ). Preprint(2012), available at http://arxiv.org/abs/1204.1328 .[28] M. Romagny, S. Wewers, Hurwitz spaces . Groupes de Galois arithm´etiques et diff´erentiels,S´emin. Congr., vol. 13, Soc. Math. France, Paris (2006), 313–341.[29] J. H. Silverman, J. Tate,
Rational Points on Elliptic Curves . Springer Verlag (1992).[30] H. Stichtenoth,
Algebraic Function Fields and Codes . Springer Verlag, GTM 254 (2008).[31] H. V¨olklein,
Groups as Galois Groups. An Introduction . Cambridge Studies in Advanced Math-ematics 53, Cambridge Univ. Press, New York (1996).
EFERENCES
Inverse Galois problem for small simple groups . Preprint (2013), available at